Transitions between irregular and rhythmic firing patterns in excitatory-inhibitory neuronal networks

J Comput Neurosci (2007) 23:217–235DOI 10.1007/s10827-007-0029-7

Transitions between irregular and rhythmic firing patternsin excitatory-inhibitory neuronal networks

Janet Best · Choongseok Park ·David Terman · Charles Wilson

Received: 24 April 2006 / Revised: 5 March 2007 / Accepted: 9 March 2007 / Published online: 16 May 2007© Springer Science + Business Media, LLC 2007

Abstract Changes in firing patterns are an importanthallmark of the functional status of neuronal networks.We apply dynamical systems methods to understandtransitions between irregular and rhythmic firing in anexcitatory-inhibitory neuronal network model. Usingthe geometric theory of singular perturbations, we sys-tematically reduce the full model to a simpler set ofequations, one that can be studied analytically. Theanalytic tools are used to understand how an excitatory-inhibitory network with a fixed architecture can gener-ate both activity patterns for possibly different valuesof the intrinsic and synaptic parameters. These resultsare applied to a recently developed model for thesubthalamopallidal network of the basal ganglia. Theresults suggest that an increase in correlated activity,corresponding to a pathological state, may be due to anincreased level of inhibition from the striatum to theinhibitory GPe cells along with an increased ability ofthe excitatory STN neurons to generate rebound bursts.

Action Editor: Carson Chow

J. Best (B) · C. Park · D. TermanDepartment of Mathematics,The Ohio State University,Columbus, OH 43210, USAe-mail: [email protected]

J. Best · D. TermanMathematical Biosciences Institute,The Ohio State University, Columbus,OH 43210, USA

C. WilsonDivision of Life Sciences,University of Texas at San Antonio,San Antonio, TX 78249, USA

Keywords Excitatory-inhibitory network ·Synchronization · Basal ganglia · Singularperturbation analysis

1 Introduction

Excitatory-inhibitory networks arise in regionsthroughout the brain including the cortex, thalamus,hippocampus, basal ganglia and the olfactory bulb.Each of these neuronal systems displays complexfiring patterns such as correlated rhythmic activity,uncorrelated spiking and propagating waves. The samenetwork may also display distinct activity patterns fordifferent values of intrinsic and synaptic parameters.For example, thalamocortical networks, responsiblefor the generation of sleep rhythms, display verydifferent rhythmic activity during different sleep stages(Steriade et al. 1993; Terman et al. 1996). Anotherexample is the basal ganglia, a group of nuclei thatplay an important role in the generation of movement.Dysfunction of the basal ganglia is associated withmovement disorders such as Parkinson’s disease andHuntington’s chorea. Numerous experiments havedemonstrated that neurons within the basal gangliadisplay a variety of dynamic behaviors; moreover,patterns of neuronal activity differ between normaland pathological states. In particular, there is a drasticincrease in correlated activity within the subthalamicnucleus and the globus pallidus in both parkinsoniananimal models and human subjects with Parkinson’sdisease (Bevan et al. 2002; Raz et al. 2000; Bergmanet al. 1994; Hurtado et al. 1999).

There have been numerous modeling and computa-tional studies of the dynamic mechanisms underlying

218 J Comput Neurosci (2007) 23:217–235

correlated rhythmic activity (Rubin and Terman 2000).These systems often display clustering in which theentire population breaks up into distinct groups orclusters; cells within a cluster are synchronized whilecells belonging to different clusters fire out-of-phase.The spindle sleep rhythm, for example, corresponds toclustered activity among neurons within the thalamus(Golomb et al. 1994; Terman et al. 1996). Previouspapers have explained how the intrinsic and synapticproperties of the excitatory and inhibitory cells interactto generate clustered activity in reduced neuronal mod-els (Rubin and Terman 2000).

Mechanisms underlying uncorrelated and/or irregu-lar spiking are, on the other hand, poorly understood.Computational modeling has been used to suggestmechanisms; however, these studies usually assume avery simple model for each neuron. It is unclear howspecific biophysical properties of the network, includingthe voltage- and time-dependent membrane conduc-tances, contribute to the emergence of irregular activ-ity. A primary goal of this paper is to suggest a possiblemechanism for how biophysical properties of neuronsmay interact to produce irregular, uncorrelated spikingin an excitatory-inhibitory network. We also demon-strate how this spiking pattern can be transformed torhythmic correlated activity by changing intrinsic andsynaptic parameters in the model.

A second goal is to develop analytic tools to studydetailed conductance-based neuronal models. Thesemodels typically involve a large number of nonlineardifferential equations that include numerous parame-ters, many of which are difficult to determine experi-mentally. One strategy for analyzing a given system isto begin with reduced equations, such as the integrate-and-fire model, for each cell. This leads to a networkmodel that is easier to analyze mathematically andimplement numerically; however, it is often difficult todeduce the precise correspondence between parame-ters in the full model and that in the simplified system;moreover, it is not clear what features of the full modelare essential in generating the firing properties. A sec-ond strategy is to begin with the full model and usedynamical systems and singular perturbation methodsto systematically reduce the full model to a simpler setof equations, one which can be studied analytically. Inthis paper, we use this second strategy to study bothrhythmic correlated activity and irregular uncorrelatedspiking arising in excitatory-inhibitory networks.

Much of our analysis is motivated by studies ofneuronal activity within the basal ganglia which havedemonstrated that there is a drastic increase in cor-related activity within the subthalamic nucleus (STN)

and the globus pallidus in both parkinsonian animalmodels and human subjects with Parkinson’s disease.The STN and the external segment of the globus pal-lidus (GPe) form an excitatory-inhibitory network. Wehave recently constructed a model for this network andcomputational simulations demonstrate that the modelcan generate rhythmic correlated activity as well as ir-regular uncorrelated spiking (Terman et al. 2002). Herewe develop analytic tools in order to understand how anexcitatory-inhibitory network, with a fixed architecture,can generate both types of activity patterns, for possiblydifferent values of the intrinsic and synaptic parame-ters. Moreover, what parameters may be responsiblefor the switch between these two activity patterns?

2 Methods—the network model

We consider an excitatory-inhibitory network in whicheach cell is represented as a single-compartmentconductance-based model. Our choice of ionic currentsis motivated by previous models of thalamic networksresponsible for the generation of sleep rhythms and amodel for the subthalamopallidal network in the basalganglia (Golomb et al. 1994; Terman et al. 2002).

Each cell satisfies a differential equation of the form

Cmv′ = −IL− IK− INa− IT − ICa− IAH P− Isyn+ Iapp

[Ca]′ = ε(−ICa − IT − kCa[Ca])

X ′ = φX(X∞(v) − X)/τX(v). (1)

Here, IL, IK, INa, IT , ICa, IAH P represent a leak, potas-sium, sodium, low threshold calcium, high thresholdcalcium and afterhyperpolarization current, respec-tively; X can be n, h or r, gating variables for thesecurrents. The currents are modeled in a standard way;the precise form as well as parameter values are givenin the Appendix and Tables 1 and 2. We note thatthe parameter values differ between the excitatory andinhibitory cells. Subscripts will be used to distinguishbetween the two types of cells; hence, vE and vI rep-resent the membrane potential of the excitatory andinhibitory cells, respectively.

In Eq. (1), Isyn represents synaptic input from othercells. The network architecture is illustrated in Fig. 1.There is synaptic coupling between the excitatory andinhibitory cell layers as well as between inhibitorycells. Let IA→B denote the synaptic current fromstructure A to a cell in structure B. Then, for ex-citatory cells, Isyn = II→E, while for inhibitory cells,

J Comput Neurosci (2007) 23:217–235 219

Table 1 Parameter values for excitatory cells

Parameter Value Parameter Value Parameter Value Parameter Value

gNa 37.5 nS VNa 55.0 mV τ 1h 500.0 ms τ 0

h 1.0 msgCa 0.5 nS VCa 140 mV τ 1

n 100.0 ms τ 0n 1.0 ms

gL 2.25 nS VL −60 mV τ 1r 17.5 ms τ 0

r 7.1 msgK 45 nS VK −80.0 mV k1 15.0 kCa 22.5gAH P 9 nS φ 0.75 φr 0.5 σn 26σh 3.0 σr 2.2 Cm 1 pF VE→I −100 mVα 5.0 ms−1 β 1.0 ms−1 θs 39.0 mV θh −39.0 mVθn −32.0 θt −63 mV θr −67 mV θm 30.0tn 1 sm 15 th 0.05 ss 2xp 1 sh 3.1 sn −8.0 ssca 8.0thn 80.0 thh 57.0 kt −7.8 kr 2.0gt 0.5 ab −30 thr −68 ε 5e−5

rth 0.25 rsig −0.07 il 0 ρ1 0.5a1 0.9 i 0 θE 0

Isyn = IE→I + II→I . In the model, the synaptic input toa cell i in structure B is given by

IA→B = gA→B(viB − VA→B)

∑

j

WijsjA.

The sum is over cells in A with the synaptic weightgiven by Wij, and each synaptic variable s j

A satisfies afirst order differential equation of the form

s′A = αA(1 − sA)H∞(vA − θA) − βAsA. (2)

H∞ is a smooth approximation of the Heaviside stepfunction.

Finally, Iapp corresponds to an applied current. Inwhat follows, we will assume that inhibitory cells re-ceive an applied current but excitatory cells do not;hence, Iapp will always represent current applied to theinhibitory cells.

3 Numerical results

3.1 Firing properties of isolated cells and networkactivity

Figure 2 illustrates the firing properties of each iso-lated cell type. The excitatory cells fire at just a fewHz. However, these cells can exhibit higher frequencyfiring either in a sustained manner when given sufficientexcitatory input or as rebound bursts following releasefrom hyperpolarizing current. The inhibitory cells canfire rapid periodic spikes with sufficient applied current.They also display bursts of activity over a range ofapplied currents.

A network of synaptically coupled E- and I- cells canexhibit several types of firing patterns, depending onnetwork parameters including the underlying architec-ture. Here we will consider both irregular, uncorrelated

Table 2 Parameter values for inhibitory cells

Parameter Value Parameter Value Parameter Value Parameter Value

gNa 120 nS VNa 55.0 mV τ 1h 0.27 ms τ 0

h 0.05 msgCa 0.1 nS VCa 120 mV τ 1

n 0.27 ms τ 0n 0.05 ms

gL 0.1 nS VL −55 mV τr 30.0 ms θs1 −35.0gK 30 nS VK −80.0 mV k1 30.0 kCa 4.0gAH P 30 nS φ 1.0 ks 2.0 σn 14.0σh −12.0 VE→I 0 Cm 1 pF VI→I −80 mVα 2.0 ms−1 β 0.04 ms−1 θs −57.0 mV θh −58.0 mVθn −50.0 gE→I 0.21 θr −70 mV θm −37.0σm 10.0 ks1 2.0 δh 0.05 δn 0.05gt 0.5 sh −12.1 sn −12.0 ε 5e−4

thn −40.0 thh −40.0 ab −20 kr −2.0


E cells

I cells

appI

Fig. 1 Network architecture. The I-cells send synaptic inhibitionboth to E-cells and to other I-cells. E-cells send excitation tothe I-cells. The I-cells receive an additional constant level ofinhibitory input

activity and more synchronous activity which we willrefer to as clustering. One of our goals is to understandhow a network with a fixed architecture can generateboth types of activity patterns, for possibly differentvalues of the intrinsic and synaptic parameters. For thenumerical simulations illustrated in Fig. 3, we consid-ered a fixed network with ten E and ten I-cells. EachE-cell sends excitation to one I-cell, while each I-cellsends inhibition to three ’nearest neighbor’ E-cells aswell as to each of the nearest neighbor I-cells. All of theparameter values used to generate Figs. 3(a) and (b) arethe same, except for IappandgI→E. Parameter values aregiven in Tables 1 and 2.

Figure 3(a) illustrates irregular activity generatedby the network. Note that there is very little correla-tion among the firing of the two shown I-cells. Thisis demonstrated by the cross-correlation also shownin Fig. 3(a). Moreover, there is very little correlationamong the spikes of a single I-cell as shown by theautocorrelation. We note that in this simulation gI→E,the strength of I → E coupling, is weak. Hence, the E-cells are to a large extent decoupled from other cells in

the network and exhibit their intrinsic periodic firing.The weak coupling does in fact create phase differencesamong the E-cells and this contributes to the overallirregular behavior of the network.

An example of a clustered rhythm is shown inFig. 3(b). The network breaks up into two clusters; cellswithin each cluster are synchronized with each otherwhile the two clusters fire out of phase. Other types ofclustered patterns are certainly possible, especially if weconsider other network architectures. The network maybreak up into more than just two clusters; moreover,membership in the clusters may be dynamic in that indi-vidual cells may fire with a given cluster for some cycles,then drop out and join another cluster. Examples ofsuch patterns will be given later.

The analysis that follows will help clarify when anetwork displays irregular activity and how a networkmay organize itself to form clustered rhythms. Thesefiring patterns depend both on network architectureand on interactions between the intrinsic and synapticproperties of cells.

3.2 Transitions and dependence on parameters

An aim of this paper is to understand how anexcitatory-inhibitory network with fixed architecturecan generate both irregular and clustered activity fordifferent parameter values. We begin by identifying pa-rameter ranges where the network has these two differ-ent behaviors. In this section, we vary two parametersin the network and observe how these parameters canaffect the activity mode of the network. The analy-sis in the following sections will serve to explain themechanisms underlying these behaviors and transitionsbetween them.

Fig. 2 Firing properties of (a)E-cells and (b) I-cells. TheE-cells fire intrinsically at afew Hz, display highfrequency firing withexcitatory input and reboundbursts following release fromhyperpolarizing current. TheI-cells fire rapid periodicspikes with sufficient appliedcurrent and may displaybursts of activity

0 500 1000 1500 2000 2500

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50

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–50

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50

time (msec)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

–100

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50

0 200 400 600 800 1000 1200 1400 1600 1800–100

–50

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50

time (msec)

V (

mV

), E

cel

l V

(m

V),

E c

ell

V (

mV

), I

cell

V (

mV

), I

cell

I = 0 I = 2

Iapp

= –.5

(a) (b)


Fig. 3 (a) Irregular firing.There is little correlationamong the firing of twodistinct I-cells or between thespikes of a single I-cell. (b)Clustered activity. Both theE- and I-cells fire in a regularmanner; the firing of twoI-cells is shown here. Thenetwork breaks up intoclusters; cells within eachcluster are synchronized witheach other while the twoclusters fire out of phase

(a) (b)

Assessing the level of synchrony in the activity of thenetwork, as well as how it depends on parameter val-ues, necessitates a quantitative measure of irregularity.Several methods for quantitatively measuring networksynchrony have been proposed and compared; theseinclude both firing time measures and continuous timemeasures.

Continuous time measures use the full voltage timeseries, thereby measuring differences in the shape ofspikes as well as in the membrane potentials of neuronsbetween firings. Two examples are χ2

V , as proposed inGolomb and Rinzel (1994), and principal componentanalysis (PCA) (Jolliffe 1986). χ2

V measures coherenceamong neurons in a network as a ratio between thetime-averaged fluctuations of the population-averagedvoltage and the population-average over each neuron’stime-averaged voltage fluctuations; thus χ2

V = 1 whenall cells have the same trajectory and χ2

V = 0 when thestate of the network is incoherent such that the time-averaged fluctuations in the population-average voltageare zero.

To perform PCA, one computes the eigenvectorsxi of the covariance matrix for the mean-adjustedvoltage traces, then sorts these principal componentsby decreasing eigenvalue. The subspace generated by(x1, ..., xm) (where m is at most the number of I-cellsin the network) has the property that projection of thedataset onto this subspace retains the largest amountof variation possible among m-dimensional subspaces.For example, if the first two principal components cap-ture nearly all of the variation in the data, then (after acertain linear transformation) two variables nearly suf-fice to describe the data. If that data was generated bythe set of I-cells in the network, this scenario stronglysuggests that the cells have separated into two clusters.

Measures based on firing times, on the other hand,ignore voltage data other than spike times. For anytwo successive firing times of a given neuron, thesemeasures map that time interval onto the unit circle,

thereby providing a natural way to associate a phase toeach firing time in the network during that interval, re-sulting in a phase vector. For an arbitrary time intervalof interest, there will in general be a large number ofsuch phase vectors, especially since one can vary thechoice of neuron for which one considers successivefiring times. Once one has a phase vector, there are avariety of methods for assigning a synchrony measureto it, including r2 defined in Kuramoto (1984) andStrogatz and Mirollo (1991) and M defined in Pinskyand Rinzel (1995). For any choice of method, onecan then take some weighted average of values overpossible choices of phase vectors to yield a synchronymeasure for the network.

We applied these four measures to a regularly-structured network of ten E-cells and ten I-cells with-out direct connections between inhibitory neurons, fordifferent values of Iapp and gI→E. Over this range of pa-rameter values, the network dynamics may be irregular,perfectly clustered, or nearly clustered; near-clusteringoccurs in this network when there are two groups ofcells that take turns firing, but the cells within a groupare not precisely synchronized.

All four measures detected general trends of irregu-lar activity and clustered firing in the network, thoughthere were minor differences. As an example, Fig. 4summarizes the results of performing PCA for net-work activity over a range of values of Iapp and gI→E.Here, variables s j

I (rather than membrane potential)are used in the PCA computation since the slowertime course of synaptic inhibition, compared to spikes,takes account of small time differences and thereby aidsdistinguishing between near-clusters and irregularity.The numbers reported indicate the number of principalcomponents required to capture 90% of the variation inthe data.

As indicated in Fig. 4, network dynamics tend to beirregular when Iapp is large but gI→E is small. Using themembrane potentials, we computed autocorrelations


0 .2 .4 .6 .8 1 1.2 1.4 1.6 1.8 2

1.5

2

2.5

1

0.5

0

–0.5

–1

–1.5

–2

–2.5

5–8

4–6

9–10

1–3

I app

EIg

Fig. 4 Dependence of clustering on the strength of inhibitionto I cells (Iapp) and I → E inhibition for a network of ten Icells (each inhibiting the three nearest E cells) and ten E cells(each contacting the closest I cell). Voltage time series of I cellswere averaged over 5 ms time windows before PCA. The legendshows the number of components required to explain 80% of thevariance for the time series

for each of the I-cells as well as cross-correlations foreach pair. The right-hand panels of Fig. 3(a) shows oneexample of autocorrelation and cross-correlation fromthe region of parameter choices predicted by the PCAto be irregular: it shows that the autocorrelation has asingle strong peak and the cross-correlation is nearlyflat when computed using the membrane potential ofI-cells in this region of Fig. 4. On the other hand, cellstend to be perfectly clustered when Iapp is small andgI→E is large. The set of parameter values resulting inirregular dynamics is larger if the network is connectedrandomly instead of regularly; it decreases in size forthe regularly-structured network as the strength of con-nections between inhibitory neurons (gI→I) increases.

In the next section, we will motivate the choice ofsome of the parameters that were not varied here, byshowing how intrinsic properties of the I-cells play anessential role in enabling irregular behavior in the net-work. While irregular activity in the network is robustin that it occurs for a large set of parameter values, itnonetheless imposes some conditions on parameters ofthe I-cells.

4 Geometric analysis of irregular activity

In this section, we will explore how irregular firingpatterns depend on the strengths of synaptic connec-tions and the intrinsic firing properties of the cells. Theanalysis does not rely on details of the E-cells; we onlyuse the fact that an E-cell fires intrinsically at a few Hz.On the other hand, details of the firing properties of themodel I-cell will be crucially important.

The analysis of irregular activity consists of severalsteps. We first consider a single I-cell. Recall that thesecells display bursting activity over a range of appliedcurrents. We will analyze this bursting activity usingfast/slow analysis to reduce the model for each I-cell toa single differential equation. We will then study howa single I-cell responds to a brief impulse of excitatoryinput. This will lead to a phase response curve (PRC)associated with the I-cell. We then consider a simplenetwork consisting of a single E-cell sending excitatoryinput to a single I-cell. Recall from Fig. 4 that we expectirregular activity if gI→E, corresponding to the strengthof synaptic input from the I to the E-cells, is weak.Here we are considering the limiting case in which thisparameter is zero. Using the PRC, we will derive a mapthat will allow us to determine precise conditions onparameters for when the periodically-forced I-cell mayexhibit irregular activity. Finally, we will comment onhow this analysis helps in the study of larger networks.

We begin by considering a single I-cell governedby system (1). Numerical simulations indicate that thecalcium concentration within an I-cell changes moreslowly than the other dependent quantities, so we willtreat CaI as the slow variable and vI, nI, hI and rI asfast variables in our fast/slow reduction. By the fastsubsystem, we mean the equations in Eq. (1) for thefast variables. The first step in the fast/slow analysis is toformally set ε = 0 in Eq. (1). If this is the case, then CaI

is constant and we may regard CaI as a bifurcation pa-rameter in the fast subsystem. The bifurcation diagramof the fast subsystem is shown in Fig. 5(a). Note that foreach value of CaI there is a unique fixed point; thesefixed points are stable for CaI > CHB and unstableat CaI < CHB. There is a subcritical Hopf bifurcationwhen CaI = CHB. A branch of unstable periodic orbitsbifurcates from the Hopf point and then “turns around”at a fold or saddle-node of limit cycles when CaI =CSN ; there exist stable limit cycles for CaI < CSN. Notethat for CHB < CaI < CSN , the fast system is bistable:there exist both a stable fixed point and a stable limitcycle.

We now include slow variable dynamics in the casewhen the I-cell bursts. The bursting solution is shownin Fig. 5(c) and the projection of the bursting trajectoryonto the bifurcation diagram is shown in Fig. 5(b).This is an example of an elliptic burster (Rinzel 1985).During the cell’s silent phase, the trajectory tracks closeto the branch of fixed points with the slow variable CaI

decreasing. Once past the Hopf point, the trajectoryjumps up to the branch of stable periodic orbits ofthe fast subsystem. The orbit then tracks close to thisbranch, generating action potentials, until it reachesthe saddle-node of limit cycles. The trajectory is then


Fig. 5 (a) Fast/slow analysisof an I-cell. Each I cell is anelliptic burster. Theprojection of a burstingsolution onto this diagram isshown in (b). Voltage andcalcium traces are shownin (c)

0.015 0.025 0.035 0.045 0.055

–80

–40

0

40

80

Calcium

V Stable Limit Cycles

UnstableFixed Points

StableFixedPoints

CHB CSN

(a)

0.015 0.025 0.035 0.045 0.055

–80

–40

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80

Calcium

V

(b)

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time

(c)

V

Ca

forced back to the branch of stable fixed points and thiscompletes one cycle of the bursting orbit.

Note that there is a delay in the silent phase fromwhen the orbit passes the Hopf point to when it jumpsup to the active phase. This is known as a “delayedbifurcation” (Nejshtadt 1985). Previous analysis hasdemonstrated that the length of delay is closely relatedto the distance between the Hopf point and where thecell jumps down (Su et al. 2003).

A very important feature of fast/slow analysis is thatone can formally reduce the full system (1) to a singleequation for just the slow variable CaI . There are twocases to consider depending on whether the cell is silentor active. Here we will simplify our qualitative analysisby using the observation that CaI changes approxi-mately linearly as it decreases during the silent phaseand also as it increases during the active phase. Thiscan be seen in Fig. 5(c). We will assume that the slowdynamics are given as

Ca′I =

{−λS when the cell is silent

λA when the cell is active.(3)

We now consider how the I-cells respond to briefexcitatory inputs and define a phase response curve(PRC) for the I-cells. We must first define a notion ofphase. We define the phase to be simply the time sincethe I-cell has last fallen down from the active to thesilent phase. Suppose, for convenience, that the I-celljumps down at time t = 0 and the total period of thecell is TI ; that is, it jumps down again at TI . Now fixt ∈ [0, TI) and suppose we stimulate the I-cell with asmall excitatory pulse at this time. The I-cell will jumpdown at some time after the pulse and we denote thistime by φ(t). This is the PRC.

Figure 6(a) shows a numerically generated PRC forthe model I-cell. Note that there exists TS ∈ (0, TI)

such that the PRC increases in an almost linear fashionfor 0 < t < TS and is nearly constant for TS < t < TI ;these features are easily understood using the geomet-ric methods discussed earlier. We assume that if thecell is silent then an impulse will force the cell into theactive phase, with the same value of the slow variable.This is a reasonable assumption, as Fig. 5(b) shows,because during much of the silent phase, the cell’strajectory is evolving close to a branch of unstable fixed

0 200 400 6000

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φ(t)

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0 400 800 12000

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π(t)

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0 100 300 5000

200

400

t

π (t)

(b)

Fig. 6 The numerically computed (plus sign) and the analyticallycomputed (solid line) PRC of a bursting I-cell (a), and the one-dimensional map associated with an I-cell periodically forcedby an E-cell when TI > TE (b) and when TE > TI (c). The

examples depicted in (b) and (c) differ in only the value of theparameter gL that controls the TE. Insets show that each clusterof points actually covers an interval


points of the fast subsystem. At the beginning of thesilent phase, the cell’s trajectory evolves near a branchof stable fixed points of the fast subsystem; however,as Fig. 5(b) suggests, during this time the trajectory isstill susceptible to being pushed away from the fixedpoints and towards the family of stable limit cycles ofthe fast subsystem. Finally, we assume that if the cell isactive, then the impulse has a negligible effect on thecell’s trajectory. Again, this assumption is reasonablebecause while in the active phase, the cell lies close toa branch of stable limit cycles of the fast subsystem. Aperturbation will push this trajectory a short distancefrom the periodic branch and the trajectory will thenquickly move back.

We note that the PRC defined here is somewhatdifferent from that used in previous studies of neuronaldynamics (Kopell and Ermentrout 2002). The PRC isusually defined for infinitesimally small perturbations.Here the perturbations cannot be too small since weassume that the perturbation will always force a silentcell to jump up to the active phase, even when the cellis at the beginning of its silent phase and lies close tostable fixed points of the fast subsystem.

Choose TS so that the unperturbed I-cell is silentfor 0 ≤ t < TS and is active for TS < t < TI . We firstconsider the case when the I-cell’s phase t satisfies0 < t < TS. From Eq. (3), the cell is a distance tλS awayfrom CSN , the jump-down value of CaI . A perturbationforces the cell to the active phase. The distance to CSN

is still tλS; however, according to Eq. (3), CaI nowincreases at rate λA until it jumps down again. It followsthat the time it takes to reach CSN is (λS/λA)t and thetime when it actually jumps down is

φ(t) =(

λS + λA

λA

)t for 0 < t < TS. (4)

Now suppose the cell is active; that is, TS < t < TI .Then the perturbation has negligible effect on the slowdynamics; hence, the cell jumps down when it would ifthere were no perturbation and

φ(t) = TI for TS < t < TI . (5)

Now consider the subnetwork consisting of just oneI-cell and one E cell, and imagine that I → E inhibitionis weak enough that we can ignore it. Since the E-cell intrinsically fires at a few Hz, it periodically sendsexcitatory input to the I-cell. We will construct a map,closely related to the PRC, that determines the result-ing firing pattern of the I-cell. The map is illustrated inFig. 7.

Let TE be the period of the E-cell, assume that theE-cell fires at time 0, fix t ∈ [0, TE) and suppose thatthe I-cell jumps down at time t. Finally, assume that the

0 100 200 300 400 500 600 700 800

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50

time

VI

VE t (t)

Fig. 7 Map associated with a periodically forced I-cell. If t is thetime from when the E-cell fires until the I-cell stops firing, thenπ(t) is the time between the termination of the next I-cell burstand the preceding time the E-cell fired

next time that the I-cell jumps down is at t0. We definethe map π : [0, TE) → [0, TE) as

π(t) ={

t0 if t0 < TE

t0 − TE if t0 > TE.

Note that there are two cases: If t0 < TE, then the I-cell jumps down before the E-cell fires again. Since theperiod of the I-cell is TI , it follows that in this case, t0 =t + TI . That is,

π(t) = t + TI if t < TE − TI . (6)

If, on the other hand, t > TE − TI , then the E-cell firesagain before the I-cell jumps down. In this case, wecompute π(t) in terms of the PRC φ. The I-cell jumpsdown at t before receiving the E-input at TE; hence, theI-cell is at phase TE − t when it receives the E-input.According to the definition of the PRC, it follows thatthe I-cell next jumps down at t + φ(TE − t). Hence,

π(t) = t + φ(TE − t) − TE if t > TE − TI . (7)

Finally, we use the explicit formula for φ given earlier,write λ for λS/λA, and conclude that

π(t) =

⎧⎪⎨

⎪⎩

t + TI if t ≤ TE − TI

t + TI − TE if TE − TI < t ≤ TE − TS

λ(TE − t) if TE − TS < t < TE.

Figure 6(b) shows a graph of this function along withsimulation results in an example with TI > TE; π(t) hasthe form shown in Fig. 6(c) when TE > TI .

In order to appreciate the behavior of the I-cellsunder periodic forcing, we must iterate the map π(t).Considering attractors of π(t) provides a means to pre-dict when periodic forcing of an I-cell may result in anirregular firing pattern: if π(t) has a chaotic attractor,we might expect the I-cell to fire irregularly. The mapπ(t) has a chaotic attractor when λ > 1, while it has astable fixed point for λ < 1. To see what the dynamicsof π(t) predict for the I-cells, we must first look more


closely at how the behavior of the I-cells differs fromthe piecewise linear approximation.

Note that for 0 < t < TS, the numerical simulationfor φ(t) appears to have a sequence of line segments;each such segment corresponds to a different integernumber of spikes. These segments appear again in thenumerical simulations for π(t), for TE − TS < t < TE.Due to the segmental nature of the numerical map, itwill not intersect the line t = π(t) for some choices ofTI , TS, TE. When the numerical map does have a fixedpoint, λ < 1 does correctly predict its stability. If thenumerical map does not have a fixed point, then λ < 1still implies that number of spikes per burst is nearlyconstant.

The condition λ > 1 is permissive of chaotic dynam-ics for the I-cells: irregular dynamics are observed inmany, but not all, such cases. In Figs. 6(b) and (c),the curves or segments comprising the attractor of π(t)suggest chaotic dynamics. Figure 8 shows a numericallycomputed return map in which the Poincare sectionis defined to be where the membrane potential, vI , ofthe I-cell is some constant. This figure corresponds toFig. 6(b). Note that the return map clearly fills up mul-tiple curves, again suggesting chaotic dynamics. Whenperiodic dynamics occur with λ > 1, the period tends tobe large and the number of spikes per burst can varyover a wide range. Whether the periodically-forced I-cell fires chaotically for a given example with λ > 1 isan intricate question involving details of the unfoldingof the saddle-node of periodic orbits bifurcation of thefast subsystem; one cannot expect a one-dimensionalmap to fully capture the complexity of the periodically-forced elliptic burster. Nonetheless, the piecewise lin-ear map π(t) makes good qualitative predictions ofthe level of complexity that will be observed in theperiodically-forced I-cell.

The picture just described holds for a single E-cell periodically forcing a single I-cell, in a parameterregime where the I-cell can burst spontaneously. Inother words, gI→E is small so that synaptic input fromthe I-cell to the E-cell is too weak to have significanteffect on the firing rate of the E-cell, and Iapp is notso strongly hyperpolarizing that I-cells need excitationto fire. As Fig. 3(a) shows, in this parameter regimethese irregular dynamics may be also observed in thefull network.

5 Geometric analysis of clustered activity

We now use geometric methods to analyze the clus-tered solutions. This will provide insights into the

0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.430.53

0.54

0.55

0.56

0.57

0.58

0.59

0.6

0.61

0.62

CaI

hI

Fig. 8 Return map for the I-cell in Fig. 6(b). The Poincare sectionis defined to be where the membrane potential, vI , of the I-cell isconstant. Note that the return map appears to fill out a curve,suggesting chaotic dynamics

possible mechanisms, both mathematical and biologi-cal, for the generation of these rhythms and how theydepend on parameters. The numerical results shownin Fig. 4 suggest that clustered activity arises if Iapp issufficiently hyperpolarizing and gI→E is large. In whatfollows, we assume that Iapp is chosen so that isolatedinhibitory cells do not fire action potentials; they can,however, fire in response to excitatory input from theE-cells. We also assume that gI→E is large enough sothat E-cells will fire bursts of action potentials whenthey are released from a sufficiently strong inhibitoryinput.

We note that mechanisms underlying clustered so-lutions consist of two separate components. The firstconcerns how two distinct clusters switch between onebeing active and the other silent. Here it is useful tothink in terms of ‘escape’ and ‘release’ as introducedin Wang and Rinzel (1992). That is, the active clustermay stop firing and this leads to release of the silentcluster from inhibition. For the ’escape’ mechanism, thesilent cluster starts firing and the resulting inhibitionsuppresses the previously active cluster. As describedbelow, the same network may exhibit multiple mecha-nisms for either escape or release depending on intrin-sic and synaptic parameters. Moreover, the distinctionbetween escape and release is often not clear-cut.

The second component concerns the behavior ofa single cluster while it is silent or active. We notethat isolated E-cells fire intrinsically at approximately3 Hz. It is perhaps not obvious why these cells fireconsiderably faster, around 20–30 Hz, while they areactive in the clustered rhythm.


Our analysis of clustered activity will proceed in sev-eral steps. We first develop a geometric understandingof firing patterns for an isolated E-cell. In particular,we demonstrate how to derive reduced equations forthese cells. This leads to rather precise conditions forwhen an E-cell will fire rebound bursts and how longthe rebound firing will last. We then consider clusteredactivity in a network consisting of two pairs of E and Icells and describe different mechanisms for escape andrelease. Finally, we propose a reduced model, basedon the geometric analysis, that is used to study firingpatterns in larger networks.

5.1 Fast and slow variables

A critical, and often nontrivial, step in analyzing amodel is to identify the fast and slow variables. Figure 9illustrates a clustered solution; it shows the membranepotential vE, the calcium concentration [Ca], the gatingvariable r of IT , and the total synaptic input Stot corre-sponding to one of the E-cells. The total synaptic inputto cell Ei is defined as

Sitot =

∑

j

WijsjI

where Wij and s jI were introduced in Eq. (2). Note

that vE is clearly the fastest variable while the cell isfiring. Moreover, [Ca] is clearly the slowest variable;[Ca] stays roughly constant at approximately .45. Inthe analysis that follows, we fix [Ca] at this constant.Remarks concerning the validity of this assumption willbe given later. In our fast/slow analysis of the clusteredsolution, we treat r and Stot as slow variables. It seemsnatural to treat r as a slow variable: the cell slowlydeinactivates while it is hyperpolarized. In order to

0 100 200 300 400 500 600 700–100

–50

0

50

0 100 200 300 400 500 600 7000

0.2

0.4

0.6

0.8

1

time

V

r Ca

Stot

(a)

(b)

Fig. 9 (a) Voltage trace of a clustered solution. (b) Calcium, theavailability r of the IT -current and the total synaptic input for thissolution

–80

–40

0

40

0 T0

Stot

V

Fig. 10 Rebound burst. The inhibitory synaptic input is on for t ∈(0, T0) and then decays at the rate βI . The membrane potentiallies at a hyperpolarized resting state and then fires a reboundburst

capture important features of the clustered solution, wewill also need to assume that the decay rate of each s j

I ,and therefore Stot is slow. We assume that the rate atwhich s j

I turns on is fast: if I j fires an action potential,then s j

I → 1 on the fast time-scale.

5.2 Geometric analysis of rebound bursts

In this section, we use geometric methods to analyzerebound bursts generated by a single E-cell followingrelease from inhibitory input. This will be an importantcomponent in the analysis of network clustered activity.Note that the rebound burst depends on many com-ponents including the strength of the inhibitory inputgI→E, the availability r of the T-current, and the decayrate of inhibition. The geometric analysis will be veryuseful in determining how rebound activity depends onthese intrinsic and synaptic parameters.

Here we consider a single E-cell, as modeled byEq. (1). We assume in this section that

II→E = gI→E Stot(vE − VI→E) (8)

where

Stot =

⎧⎪⎨

⎪⎩

0 if t < 0

1 if 0 ≤ t ≤ T0

e−βI t if T0 < t.

(9)

That is, we assume that the E-cell receives inhibitorysynaptic input for 0 ≤ t ≤ T0 and then this input decaysat the rate βI for t > T0. Figure 10 shows a solutionof these equations. The cell lies at a hyperpolarizedmembrane potential while it receives inhibitory inputand then fires a burst of action potentials once it isreleased from inhibition.

The first step in the fast/slow analysis is to computethe bifurcation diagram of the fast subsystem with theslow variables considered as parameters. Note thatthere are three fast variables, namely vE, h and n. Theslow variables are Stot and r. We are assuming that [Ca]


–5 –4 –3 –2 –1 0 1 2

–80

–60

–40

–20

0

20

40

σ

VLimit Cycles

Fixed Points

Hopf point

SNIC

(a)

0 1 2 0

.5

1

σ

∑

gI –> E

r

r*

σ*

(b)

Fig. 11 (a) Bifurcation diagram of the fast subsystem corre-sponding to a rebound burst. Here r = .5 and the bifurcationparameter is σ . The branch of fixed points forms a Z -shapedcurve. There is a branch of limit cycles that originate at a sub-

critical Hopf point and terminate at a SNIC. (b) Two-parameterbifurcation diagram. The curve � corresponds to the SNIC pointsand it separates the regions of spiking and quiescent behavior ofthe fast subsystem

is constant, as described in the preceding section. Inwhat follows, we let

σ = gI→E Stot (10)

and treat (σ, r) as the bifurcation parameters in thefast subsystem. Using σ instead of Stot will simplify thecalculations that follow. We need to identify the regionin the slow (σ, r) phase-space where the fast subsystemexhibits oscillations. We first fix r and consider σ to be asingle bifurcation parameter. The bifurcation diagramwith r = .5 is shown in Fig. 11(a). Note that there is aZ -shaped curve of fixed points; those along the lowerbranch are stable with respect to the fast subsystem.We denote the position of the left ‘knee’ of this curveas σ0(r). There is a subcritical Hopf bifurcation atσ ≈ −1.5 from which there bifurcates a branch of pe-riodic orbits. This branch terminates at an orbit homo-clinic to the left knee at σ0(r). The homoclinic orbitcorresponds to a saddle-node on an invariant circle(SNIC).

This bifurcation structure persists for each r, 0 ≤r ≤ 1. The two-parameter bifurcation diagram is shownin Fig. 11(b). The curve � ≡ {σ = σ0(r)} correspondsto both the left knees and homoclinic (SNIC) pointsof the one-parameter bifurcation diagrams with r fixed.The fast subsystem exhibits sustained oscillations forσ < σ0(r) and a stable resting state for σ > σ0(r). Notethat σ0(0) < 0. Hence, if r is sufficiently small thenthe fast system does not exhibit oscillations for σ > 0.Let r∗ be such that σ0(r∗) = 0. In the spiking region,the frequency of the spikes depends on both σ and r.We denote the frequency as �(σ, r); it is an increasingfunction of r and a decreasing function of σ . Moreover,the frequency approaches zero near �, the curve ofhomoclinics.

We now treat r and σ as dynamic variables in the fullE-cell model. The projection of the rebound burst ontothe (σ, r) slow phase-plane is shown in Fig. 11(b). Whilethe E-cell receives inhibition, σ = gI→E and r increasesto an elevated level. When inhibition is turned off, σ

decreases and spiking begins shortly after (σ, r) crosses�. While the cell is spiking, r decreases and the slowvariables (σ, r) must eventually cross � again. At thispoint, spiking is terminated. We note that (σ, r) leavesthe firing region near σ = 0 and, therefore, near r = r∗.

Geometric analysis leads to a deeper understandingof how the rebound burst depends on parameters. Inorder to analytically estimate the duration of the re-bound burst, we make some simplifying assumptions.We will assume that τr(v) is piecewise constant; that is,τr(v) = τS, a constant, while the cell is hyperpolarizedand τr(v) = τA, another constant, while the cell is firingaction potentials. We further assume that r∞(v) = 0while the cell is active.

We need to estimate the resting potential of the cellwhile it receives inhibitory input. Note that at hyperpo-larizing potentials,

IK = INa = IT = I[Ca] ≈ 0.

In this case, we can set the right hand side of the firstequation in Eq. (1) to be zero and then compute theresting potential. To simplify the notation, we let

C0 = [Ca][Ca] + k1

.

Then the resting potential is given by

VR = gLVL + gAH PC0VK + gI→E StotVI→E

gL + gAH PC0 + gI→E Stot. (11)


Together with the assumptions given above, it followsthat

r′ ={

(r∞(VR)−r)/τS while the cell receives inhibition

− rτ−A while the cell is spiking.

We now estimate the duration of the rebound burst.The cell receives inhibitory inhibit for 0 < t < T0. Dur-ing this time,

r′ = (r∞(VR) − r)/τS.

Assuming that r(0) = 0, it follows that

r(T0) = r∞(VR)(1 − e−T0/τS). (12)

Once the cell is released from inhibition, there is adelay until (σ, r) crosses � and the cell begins to spike.We now consider two separate cases which allow us toexplicitly compute the duration of the rebound burst.First we assume that the rate of decay of Stot is fast– that is, βI is large – compared to τS, the rate atwhich r turns on. This implies that the delay in firingis negligible. Moreover, after the cell is released frominhibition, (σ, r) evolves horizontally until (σ, r) crosses� and then r decays slowly at the rate τA. Burstingactivity persists until (σ, r) leaves the spiking region atapproximately r = r∗, as shown in Fig. 11. In this case,

r(t) = r(T0)e−(t−T0)/τA

for t > T0. Choosing t∗ so that r(t∗) = r∗, we find thatthe duration of the rebound burst is given by

TRBD = t∗ − T0 = τAln(

r∞(VR)

r∗(1 − e−T0/τA)

). (13)

For the second case, we do not assume that βI isnecessarily large; however, we assume that r∞(v) ≡ 1while the cell receives inhibitory input. We note thatthe curve σ0(r) is nearly constant for r sufficientlylarge. Here we assume that σ0(r) = σ∗, a constant, forr sufficiently large; moreover, when the trajectory (σ, r)crosses �, it does so for σ = σ∗. It is then straightfor-ward to compute the time at which (σ, r) enters thefiring region. We denote this time as T1. From Eq. (9),it follows that

Stot(t) = e−βI(t−T0) for t > T0.

Hence,

σ(t) = gI→Ee−βI(t−T0) for t > T0.

Now (σ, r) enters the firing region when σ(T1) = σ∗.Therefore,

T1 = T0 + 1

βIln

gI→E

σ∗. (14)

For 0 < t < T1, r satisfies r′ = (1 − r)/τS with r(0) = 0.

Hence,

r(T1) = 1 − e−T1/τS = 1 −(

σ∗gI→E

) 1βI τS

e−T0/τs (15)

As before, bursting activity persists until (σ, r) leavesthe spiking region at approximately r = r∗. Since,

r(t) = r(T1)e−(t−T1)/τA

for t > T1, it follows that

T2 − T1 = τAlnr(T1)

r∗(16)

and the duration of the bursting activity is given by

TRBD = (T2 − T1) + (T1 − T0)

= τAlnr(T1)

r∗+ 1

βIln

gI→E

σ∗. (17)

We note that Eqs. (13) and (17) depend on twoconstants, r∗ and σ∗ that do not appear in the modelequations. These are, however, easy to compute usingbifurcation software such as XPPAUT (Ermentrout2002). We remark that r∗ and σ∗ do not depend on theother parameters that appear in these formulas.

In Fig. 12, we plot the duration of bursting activityas a function of T0, the amount of time that the cellreceives inhibitory input. The solid curves representthe predicted duration, given by either Eqs. (13) or(17), and the circles represent values computed fromsolutions of the model equations. For Fig. 12(a), βI = 1is rather large and Eq. (13) was used to compute theburst duration. In Fig. 12(b), βI = .1 is smaller; in thiscase, we used Eq. (17). The analysis predicts that theburst duration is an increasing function of T0. Increas-ing T0 allows r to reach a higher level before the cellis released from inhibition. This, in turn, allows (σ, r) tospend more time in the spiking region, thus generating alonger burst. Note, however, that the curves generatedby solutions of the model equation are discontinuous;moreover, these curves actually decrease slightly be-tween discontinuities. The discontinuities correspondto values of T0 where there is a change in the numberof spikes in the rebound activity. As T0 increases, thenumber of spikes may also increase; the addition of onespike adds approximately 40 ms to the burst duration.In order to understand why the duration decreasesslightly, we must consider how the frequency of spikeswithin a burst changes with respect to T0. If we increaseT0, then r reaches a higher level while it receives in-hibitory inhibit. It then follows that (σ, r) enters thespiking region at a higher value of r. Now recall that


Fig. 12 The duration ofbursting activity as a functionof T0. The solid curvesrepresent the predictedduration and the circlesrepresent values computedfrom solutions of the modelequations

0

50

100

150

0 40 80 120 160 2000

50

100

150

T0

TRBD

TRBD

(a)

0

40

80

120

0 40 80 120 160 2000

40

80

120

T0

(b)

TRBD

TRBD

the frequency of spikes �(σ, r) is an increasing functionof r. Hence, larger values of T0 lead to faster spikes. Ifthe number of spikes per burst does not change, thenthis results in a shorter burst duration.

5.3 Fast/slow description of clustered activity

We now describe the geometric construction of clus-tered solutions arising in the network shown in Fig. 13consisting of two pairs of E and I cells. Note that eachEi excites just one of the I-cells, namely Ii. However,each I-cell may send inhibition to both E-cells butwith different synaptic weights, denoted as Wa and Wb .Hence, the synaptic current to Ei is given by

IiI→E = gI→E

(Wasi

I + Wb s jI

) (vi

E − VI→E)

≡ σi(vi

E − VI→E). (18)

We assume that Iapp is sufficiently strong so that iso-lated I-cells do not fire intrinsically; however, they canfire in response to excitatory inputs. We continue toassume that [Ca] is constant. The issue of how to choose[Ca] will be addressed in the next section.

We consider (σi, ri) to be slow variables. Then the bi-furcation diagram of the fast subsystem correspondingto each E-cell is precisely as described in the previoussection.

Fig. 13 Excitatory-inhibitorynetwork. The synaptic weightfrom Ii to E j is Wa if i = jand Wb if i = j

I1

I2

E1

E2

Wa W

a

Wb

To simplify the discussion, we first assume thatWa = 0; that is, cell Ii does not send inhibition backto Ei. In this case,

σ1 = gI→EWb s2I and σ2 = gI→EWb s1

I .

A clustered solution, for this case, is illustrated inFig. 14. The first row shows the v-profiles of each E-cell and the second and third rows show the r- and σ -profiles. The projections of these slow variables ontothe slow (σ, r) phase plane is shown in Fig. 15.

We denote the two clusters as Cluster 1, consisting ofE1 and I1, and Cluster 2, consisting of E2 and I2. Westep through the solution starting at the end of Cluster1’s active phase until the end of Cluster 2’s active phase.This will correspond to one-half of a complete cycle ofthe clustered solution. The second half is the same, butwith the roles of Cluster 1 and Cluster 2 reversed.

–100

0

0

0.5

1

0 100 200 300 4000

1

2

time

σ

r

V

(a)

(b)

(c)

Fig. 14 A clustered solution showing (a) the E-cells’ membranepotentials along with the r-profiles (b) and the σ profiles (c)


0 1 2 0

0.5

1

σ

∑

r C2( t

0 )C

2( t

1 )

C1( t

0 )

C1( t

1)

r0

Fig. 15 Projection of the clustered solution onto the slow phaseplane. The positions of Cluster 1 and Cluster 2 are given by C1(t)and C2(t), respectively

Choose t0 so that Cluster 1 fires the last spike ofits active phase when t = t0. We will assume that, atthis time, r1(t0) = r0 where r0 needs to be determinedfrom the analysis. Note that s1

I(t0) ≈ 1 and, therefore,σ2 ≈ gI→EWb . This is because Cluster 1 fires a spikeat t = t0 and the synaptic variables activate on the fasttime-scale. We further note that r2(t0) is at an elevatedlevel, as shown in Fig. 15. This is because E2 had beenreceiving inhibitory input from I1 during Cluster 1’sactive phase.

Once Cluster 1 stops firing, E2 is released frominhibition; that is, σ2 decreases. There is a delay untilthe slow variables (σ2, r2) cross � and enter the spikingregion. Suppose that this happens when t = t1. Dur-ing this delay, r1 continues to decrease. Once (σ2, r2)

crosses �, Cluster 2 begins to spike. During each spike,s2

I → 1 so that σ1 → gI→EWb . In between spikes, s2I and

therefore σ1 decay. If the interspike interval is suffi-ciently long, then it is possible for (σ1, r1) to cross � intothe spiking region before Cluster 2 is able to generateanother spike. If this is the case, then Cluster 1 will haveescaped from inhibition, thus terminating Cluster 2’sactive phase. This is, in fact, what happens for the so-lution shown in Fig. 14. Recall that the spike frequency�(σ, r) is an increasing function of r. If r is sufficientlysmall, then the frequency may be small enough to allowfor the silent cluster to escape from inhibition. In whatfollows, We choose t2 so that Cluster 2 fires its last spikewhen t = t2. If r2(t2) = r0, then the roles of Cluster 1and Cluster 2 are reversed; this represents one half ofa complete cycle of the clustered solution. We now useanalysis similar to that given in the preceding section

to estimate the period of the clustered solution. Herewe make the same assumption as before concerningτr(v); moreover, we assume that r∞(v) = 1 if an E-cell receives inhibitory input and r∞(v) = 0 if it doesnot. We further assume, as before, that � is a verticalcurve with σ0(r) = σ∗ for r sufficiently large and theslow variables cross into the spiking region at σ = σ∗.

Let t0, t1 and t2 be as described above. Suppose thatr2(t0) = R0 where R0 needs to be determined from theanalysis. For t0 < t < t1, σ2 satisfies σ2

′ = −βIσ2. Sinceσ2(t0) = gI→EWb and σ2(t1) = σ∗, it follows that

t1 − t0 = 1

βIln

gI→EWb

σ∗. (19)

Moreover, r2 satisfies r′2 = (1 − r2)/τS for t0 < t < t1.

Hence,

r2(t1) = 1 + (R0 − 1)e−(t1−t0)/τS

= 1 + (R0 − 1)

(σ∗

gI→EWb

)1/βIτs

(20)

Now, r′2 = −r2/τA for t1 < t < t2 and r2(t2) = r0. It

follows that

t2 − t1 = τAlnr2(t1)

r0. (21)

We next consider r1. Note that r1(t0) = r0 and

r′1 =

{− r1/τA for t0 < t < t1(1 − r1)/τS for t1 < t < t2.

(22)

Hence,

r1(t1)=r0e−(t1−t0)/τA and r1(t2)=1+(r1(t1)−1)e−(t2−t1)/τS .

Setting r1(t2) = R0 leads to an equation for R0. Thisequation is easier to write if let

ρ = 1 − R0 and κ = σ∗gI→EWb

.

Then ρ satisfies

ρ(1 − ρκ1/τsβI

)τA/τS = (1 − r0κ1/τAβI )rτA/τS

0 . (23)

Once we solve for ρ, the total period is given by

t2 − t0 = (t2 − t1) + (t1 − t0)

= τAln(1 − ρκ1/τSβI

r0) − 1

βIln κ. (24)

To complete the analysis, we must compute r0. Weneed to choose r0 so that if the active cluster fires aspike at r = r0, then the silent cluster is able to escape.Suppose that this spike occurs when (σ, r) = (σ0, r0).Then the time until its next spike is the reciprocalof the frequency �(σ0, r0). Note that σ0 = σ∗e−βI(t2−t1).Moreover, the time it takes the silent cluster to escape


Fig. 16 The half-period ofclustered solutions using boththe model equations and theanalytic formula

1 2 3 4100

150

200

250

300

Hal

f Pe

riod

1 2 3 450

100

150

200

250

1 2 3 450

100

150

200

gI → E

gI → E

Hal

f Pe

riod

1 2 3 40

50

100

150

200

βI = .04

βI = .03β

I = .02

βI = .05

is t1 − t0; this is given in Eq. (19). Therefore, we definer0 according to the formula:

�(σ0, r0) = βI

{ln

gI→EWb

σ∗

}−1

(25)

We have computed the half-period of clustered solu-tions using both the model equations and the analyticformula. The results shown in Fig. 16 demonstrate thatthe periods match quite well over a wide range of valuesof gI→E and for different values of β. To derive theanalytic formula, we needed to estimate the firing rate�(σ, r), since it appears in Eq. (25). This was done usingXPPAUT.

5.4 Other types of solutions

The network shown in Fig. 13 may exhibit other typesof solutions besides clustering. For example, there mayexist a suppressed solution, in which one cell contin-

uously spikes and the other cell remains silent. Sucha solution is shown in Fig. 17(a) and the projectionof this solution onto the slow phase-plane is shown inFig. 17(b). Here Wa > 0, so that each Ei both excitesand receives inhibition from Ii. Suppose that Cluster1 is the active cluster. When E1 fires a spike, thisinduces I1 to fire; this, in turn, causes σ1 → gI→EWa andσ2 → gI→EWb . After the spike, both σ1 and σ2 decay.If (σ1, r1) enters the firing region before (σ2, r2), thenCluster 1 will spike again. Note that in between thespikes, r1 first increases while (σ1, r1) is in the silentregion and then r1 decreases once (σ1, r1) enters thespiking region before Cluster 1 actually spikes. If theincrease and then decrease in r1 balance, then Cluster1 exhibits sustained oscillations and a suppressed solu-tion results.

We have so far assumed that the I-cells always re-spond to a spike from an E-cell. However, this may notnecessarily be the case if Iapp is too hyperpolarizing or

Fig. 17 Suppressed solution.One cell spikes periodicallywhile the other cell is alwaysquiet (a). The projection ofthis solution onto the slowphase plane is shown in (b)

0 200 400 600 800 1000

–50

0

50

0 200 400 600 800 1000

–50

0

50

V1

V2

(a)

time0 1 2

0

0.2

0.4

0.6

.8

1

σ

rCluster 1

Cluster 2

FiringRegion

(b)


0 250 500–80

0

60

0 250 500–100

0

40

t

vI

vE

Fig. 18 A clustered solution in which the termination of spikingis due to the build-up of the IAH P-current in the I-cells. Note thatthe I-cell does not respond to the last E-cell’s spike

gE→I is too small. How an I-cell responds also dependson its outward IAH P current. It is possible that the I-cell will not respond to excitation if IAH P is too strong.Figure 18 shows a clustered solution in which the switchbetween silent and active clusters depends on the IAH P-current in the I-cells. While an active I-cell fires, itscalcium level increases; this results in an increase ofIAH P, making that cell less sensitive to excitation fromthe active E-cell. Once calcium is sufficiently high,so that IAH P is sufficiently strong, the I-cell can nolonger respond to excitation; this then terminates thecluster’s active phase. Once the I-cell stops firing, thesilent cluster is able to rebound from inhibition and thiscluster ’takes over’.

5.5 A reduced model for larger networks

We have, so far, considered small networks. Here weconsider arbitrary excitatory-inhibitory networks anddemonstrate how the geometric analysis can be used toreduce the full model to a simpler set of equations. Inearlier sections, we assumed that the calcium concen-trations [Ca]i of the E-cells are constant. We will notmake this assumption here. Instead, we will treat the[Ca]i as slow variables.

Suppose that the number of E− and I-cells is NE andNI , respectively. We will assume that each I-cell fires aspike if and only if it has just received excitatory inputfrom an E-cell. For 1 ≤ i ≤ NE, let

σi = gI→E

∑

j

WijsjI (26)

where the various terms were defined in Eq. (2).We need to consider the (σ, r) phase plane as shown

in Fig. 11. Let F and S denote the firing and silentregions, respectively. We note that these regions nowdepend on the calcium concentration [Ca]; this wasconstant in earlier sections. The firing rate of a cellwhile in the firing region also depends on [Ca]. For thisreason, we now denote the firing rate as �(σ, r, [Ca]).

There are three sets of slow variables. These are: (1)ri, the availability of the T-current in the E-cells, (2)sI

j , the synaptic variables of the I-cells, and (3) [Ca]i,the calcium concentrations of the E-cells. We nowconstruct reduced equations for each of these variables.

We first consider the [Ca]i. We assume that [Ca]i in-creases by an amount � each time that Ei fires an actionpotential; moreover, [Ca]i decays at the exponentialrate Kca between Ei-spikes. Here, � is a constant thatneeds to be determined and KCa = εkca where ε andkca were defined in Eq. (1). We note that it is easy toestimate � by considering a single E-cell and observingthe change in [Ca] after an action potential. Then [Ca]i

satisfies the differential equation

[Ca]′i = �∑

k

δ(tk) − Kca[Ca]i

where δ(t) is the Dirac-delta function and {tk} is theset of times that Ei spikes. If we integrate this equa-tion, then we formally obtain the following equationfor [Ca]i:

[Ca]′i = ��(σi, ri, [Ca]i) − Kca[Ca]i. (27)

We next consider ri. We make the same simplifyingassumptions concerning τr as before and we assumethat r∞(vE) = 0 whenever Ei is spiking. Then each ri

satisfies

r′i =

{(r∞(VR) − ri)/τS if (σi, ri) ∈ S([Ca]i)

− ri/τA if (σi, ri) ∈ F([Ca]i).

Here, VR is given in Eq. (11).Finally, we consider s j

I . Here we will assume that s jI

increases by the amount αI(1 − s jI) each time that I j

fires an action potential and s j decays at the exponential

(a2)(a1) (b1) (b2)

Fig. 19 Dynamic clustering in the full (a) and reduced (b) model.Each panel has six columns and each column represents a neu-ron’s activity as it evolves over time (2,000 ms). Solutions of thefull (respectively, reduced) model alternate between the patternsshown in (a1) and (b2) (respectively, (b1) and (b2)). Differentsubpopulations (clusters) of cells take turn firing together, andthe membership of each cluster changes with time


Fig. 20 Dynamic clustering in the (a) full and (b) reduced mod-els. There are twenty pairs of E- and I-cells. Each E-cell excitesone I-cell and each I-cell inhibits three E-cells

rate βI between I j-spikes. Then, preceding as before, s jI

satisfies the equation

s jI

′ = αi(1 − s jI)

∑�(σi, ri, [Ca]i) − βIs j

I (28)

where the sum is taken over all i such that there existsan Ei → I j connection.

The reduced model is very easy to implement nu-merically. We illustrate this with two examples. Thesolution shown in Fig. 19 can be described as “dynamicclustering”: different subpopulations (clusters) of cellstake turns firing together and the membership of eachcluster changes with time. For this simulation, therewere six pairs of E− and I-cells. Each panel has sixcolumns and each column represents an E-cell’s activityas it evolves over time. Note in Fig. 19(a1), cell 1 takesturns firing with either cell 2 or cell 5. After some time,this pattern changes and cell 1 takes turns firing withcell 3 and cell 6. This is shown in Fig. 19(a2). Over time,the network switches between the dynamics shown inFigs. 19(a1) and (a2). For Figs. 19(b1) and (b2), weshow solutions of the reduced model. Note that thereduced model exhibits dynamics very similar to thatof the full model.

As another example, consider the solution illustratedin Fig. 20 in which there is weak clustering. We simu-lated the reduced model with exactly the same architec-ture as that used to generate Fig. 20(a) and the result isshown in Fig. 20(b).

6 Discussion

We have demonstrated that excitatory-inhibitory net-works can generate both irregular, uncorrelated firingpatterns and rhythmic, correlated activity. These pat-terns may exist in a network with a fixed architecturebut for different values of the intrinsic and synaptic pa-rameters. The mathematical analysis presented in thispaper allows us to characterize for which parameters a

particular pattern exists and which parameters may beresponsible for switching between different patterns.

The primary strategy for analyzing the detailedmodel is to use geometric dynamical systems andsingular perturbation methods to reduce the model to asimpler set of equations. The first step is to identify fastand slow processes. Then the reduced model consistsof equations for just the slow variables. We note thatevery parameter in the full model must play a role in thereduced model; hence, analysis of the reduced modelleads to an understanding of how each parameter inthe original model shapes network behavior. While it isobvious where parameters in the slow equations appearin the reduced model, it is sometimes not obvious whatrole parameters in the fast equations play. Parametersin the fast equations may appear in the reduced modelin two places. The first is when we solve for the fast vari-ables in terms of the slow variables; see, for example,Eq. (11). The second involves the bifurcation structureof the fast subsystem. For example, in the analysisof irregular activity, parameters in the fast equationsdetermine the location of the bifurcation points CSN

and CHB; see Fig. 5. In the analysis of clustered activity,parameters in the fast equations determine the locationof the curve �, shown in Fig. 11, which separates thequiescent and the spiking regions.

The analysis helps to identify which features of themodel are essential for producing a given pattern andwhich are superfluous. For clustered activity, the I-cellsfire only in response to excitatory input. Any neuronalmodel for the I-cells that satisfies this property wouldhave sufficed, not just the detailed model consideredhere. Another requirement for clustered activity is thatthe E-cells generate robust rebound bursts. We notethat clustered activity has been analyzed in several ear-lier studies of neuronal models using geometric meth-ods similar to those employed here. Most of these ear-lier studies considered simplified models which did notinclude the spike-generating mechanisms. One novelfeature of the analysis presented here is that it illus-trates how to extend the previous analysis to includespiking activity.

Irregular activity depends on several properties ofthe network. First, the I-cells have the bifurcationstructure of an elliptic burster. In particular, the fastsubsystem exhibits bistability with stable resting andstable oscillatory behavior over a range of the slowvariable. We note that this property is shared by othertypes of bursters, including square-wave bursting. Inthe construction of the phase response curve, we as-sumed that if the I-cell is silent, then an impulse willforce the cell into the active phase. This property ismore robust for elliptic bursters than for square-wave


bursters because for elliptic bursters, the cell’s trajec-tory lies very close to an unstable fixed point duringmuch of the silent phase. In the model considered here,the slow variable corresponds to intracellular calcium.Other models for elliptic or square-wave bursting in-volve other slow processes such as the inactivation of apersistent sodium current or activation of a potassiumA-current. The analysis presented here demonstrateshow these currents may contribute to irregular spiking.

Other properties must be satisfied in order for thenetwork to exhibit irregular and uncorrelated activity.We assumed that inhibitory input to the E-cells is weak.Hence, the primary role of the E-cells is to periodicallyreset the I-cells. Finally, irregular activity only emergesif certain relationships between the intrinsic propertiesof the cells are satisfied. For example, the analysis inSection 4 demonstrates that in order for there to existirregular spiking, we must have that λS > λA where λS

and λA correspond to the rates at which the I-cell’sslow variable evolves while that cell is silent and active,respectively.

The types of firing patterns described in this pa-per arise in many neuronal systems. Irregular spikinghas been reported in, for example, cortical neuronsand hippocampal inhibitory cells (van Vreeswijk andSompolinsky 1998; Parra et al. 1998) . The origins of thisirregularity is not well understood and computationalmodeling has been used to suggest mechanisms under-lying this activity. One such hypothesis is that the irreg-ularity is due to a balance of excitatory and inhibitorycurrents. Theoretical studies have demonstrated thatsuch a balanced network can produce a highly irregularstate with low levels of synchrony if the total excitatoryand inhibitory currents as well as the total current fromoutside the network into a cell are large compared tothe neuronal threshold. These studies assumed a verysimple model for each neuron; often, each neuron isdescribed as a binary variable or as a leaky integrate-and-fire cell. It is unclear how specific biophysicalproperties of the network, including the voltage- andtime-dependent membrane conductances, contribute tothe emergence of irregular or some other type of activ-ity. The analysis presented here suggests one possiblemechanism for how biophysical properties of neuronsmay interact to produce irregular dynamics.

The subthalamic nucleus (STN) and external seg-ment of the globus pallidus (GPe) form an excitatory-inhibitory network within the basal ganglia (Termanet al. 2002). Experiments have demonstrated that pat-terns of neuronal activity within these nuclei change be-tween a normal and a pathological state. In particular,there is a drastic increase in correlated activity withinthese nuclei in both parkinsonian animal models and

human subjects with Parkinson’s disease (Bevan et al.2002; Raz et al. 2000; Bergman et al. 1994; Hurtadoet al. 1999). The origin of each of these activity pat-terns is unclear. Although the cause of the transitionfrom normal to pathological activity is also unknown, ithas been established that this change correlates with aloss of the neurotransmitter dopamine within the basalganglia. The results in this paper suggest that the switchbetween normal and pathological activity may be dueto an increased level of inhibition from the striatumto the GPe along with an increased ability of the STNneurons to generate rebound bursts. The later propertyis consistent with experiments that demonstrate thatapplication of dopamine to STN neurons reduces theirresponse to inhibitory inputs (Cragg et al. 2004).

Acknowledgements This work was partially funded by the NSFunder agreement 0112050 and grant DMS0514356 (DT) and byNINDS grant NS047085 (CJW).

Appendix

Equations (1) were introduced in Terman et al. (2002).The leak current is given by IL = gL(v − VL), andthe other voltage-dependent currents are described bythe formalism of Hodgkin and Huxley (1952) as fol-lows: IK = gKn4(v − VK), INa = gNam3∞(v)h(v − VNa),and ICa = gCas2∞(v)(v − VCa. The T current in theE-cell (STN cell) is modeled as IT = gTa3∞(v)b∞(r)2

(v − VCa) and in the I-cell (GPe cell) is modeled inthe simpler form IT = gTa3∞(v)r(v − VCa). In Eq. (1),for x ∈ {m, h, n, s, a, r}, the function x∞(v) takes theform x∞(v) = {1 + exp[(v − θx)/σx]}−1, and for x ∈{h, n, r}, the function τx(v) takes the form τx(v) =τx/ cosh[(v − θx)/2σx]. For the T current inactivationvariable b , we used b∞(r) = 1/[1 + exp[(r − θb )/σb ]] −1/[1 + exp(−θb/σb )]. The unusual way of modeling Tcurrent inactivation here was designed to combine theeffects of a hyperpolarization-inactivated inward (sag,or H) current with those of a T current, making therebound bursts of an STN cell more prominent. Theparameter values used in these equations are listed inTable 1 and Table 2 for E-cells and for I-cells, respec-tively. Numerical simulations were performed usingXPPAUT (Ermentrout 2002).

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Transitions between irregular and rhythmic firing patterns in excitatory-inhibitory neuronal networks

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